Kinematics

JEE Main level, single correct, medium

226 of 308

Question 226 of 308

Two projectiles AA and BB are thrown with initial velocities of 40ms−140m{s}^{-1} and 60ms−160m{s}^{-1} at angles 30∘30^{\circ} and 60∘60^{\circ} with the horizontal respectively. The ratio of their ranges respectively is (g=10ms−2)(g=10m{s}^{-2})

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Answer: (A) 4:94:9

The range (R1)\left({R}_{1}\right) of the first projectile is given by
R1=u12sin⁡2θ1g...(1){R}_{1}=\dfrac{{{u}_{1}}^{2}\sin 2{\theta }_{1}}{g}...(1)
The range (R2)\left({R}_{2}\right) of the second projectile is given by
R2=u22sin⁡2θ2g...(2){R}_{2}=\dfrac{{{u}_{2}}^{2}\sin 2{\theta }_{2}}{g}...(2)
Divide equation (1) by equation (2) and simplify to obtain the ratio of the ranges.
R1R2=u12sin⁡2θ1gu22sin⁡2θ2g=u12sin⁡2θ1u22sin⁡2θ2...(3)\begin{matrix}\dfrac{{R}_{1}}{{R}_{2}} & = & \dfrac{\dfrac{{{u}_{1}}^{2}\sin 2{\theta }_{1}}{g}}{\dfrac{{{u}_{2}}^{2}\sin 2{\theta }_{2}}{g}} \\ & = & \dfrac{{{u}_{1}}^{2}\sin 2{\theta }_{1}}{{{u}_{2}}^{2}\sin 2{\theta }_{2}}...\left(3\right)\end{matrix}
Substitute the values of the known parameters into equation (3) to calculate the required ratio.
R1R2=402×sin⁡60∘602×sin⁡120∘=49\begin{matrix}\dfrac{{R}_{1}}{{R}_{2}} & = & \dfrac{{40}^{2}\times \sin {60}^{^{\circ}}}{{60}^{2}\times \sin {120}^{^{\circ}}} \\ & = & \dfrac{4}{9}\end{matrix}

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